[aspect-devel] Internal heating in aspect (Ludovic Jeanniot)

Max Rudolph maxrudolph at ucdavis.edu
Wed Aug 29 12:49:21 PDT 2018 

Rene,
Thank you for summarizing the current state of affairs. I would suggest
that when you make the changes in (1)-(2), you also make ASPECT print at
least a warning when the entropy viscosity exceeds, say, 1% of the thermal
conductivity. It might be even better for the code to exit in with an error
unless a parameter is explicitly set to ignore such a condition.

It also seems that one way around (1) is to use DG elements for
composition. Is this what most users are doing anyways?

Thank you for addressing these issues so quickly. I am happy to help with
testing as needed.

Best,
Max

On Wed, Aug 29, 2018 at 12:16 PM Rene Gassmoeller <
rene.gassmoeller at mailbox.org> wrote:

> Hi all,
>
> I think by now we have a pretty good understanding of the problem, however
> there is no clear path forward yet. I have done some further testing with
> the shell_simple_3d cookbook:
>
>  - As Max pointed out the artificial viscosity at resolution <= 2 global
> refinement in spherical models is bigger than the natural diffusion,
> although it reduces drastically with resolution (compare to a natural
> conductivity of ~4 W/m*K).
>
> Global refinement / Maximum artificial viscosity timestep 0 / timestep 2:
>
> 1 (4 radial elements): 94.1 W/m*K  /  59.9 W/m*K
>
> 2 (8 radial elements): 48.9 W/m*K / 6.12 W/m*K
>
> 3 (16 radial elements): 24.8 W/m*K / 0.92 W/m*K
>
> There are some pitfalls here, in particular the artificial viscosity in
> the output of timestep 0 is in general much bigger than in later timesteps
> (and it only scales linearly with cell size), because we do not have a
> velocity solution yet. Timestep 1 is somewhat unreliable as well, because
> the BDF2 scheme is not fully initialized. Nevertheless, for later timesteps
> the artificial viscosity scales at least with cell size squared (h^2) as
> expected (actually slightly better, because it is not only h^2, but also
> based on the residual of the equation, which reduces as well, ideally with
> h^3). This means higher resolutions should significantly reduce the total
> amount of artificial diffusion, although it might still be significant (>1%
> of total diffusion) up to resolution 4 or so, which seems unacceptably
> expensive. Moreover, as Scott and Cedric mentioned there will always be an
> imbalance between the heat fluxes at the top and bottom boundary with this
> method, unless we use radially uniformly spaced elements, or disable the
> stabilization, although the magnitude of the imbalance should reduce with
> increased resolution. Also the thermal conductivity within the domain will
> be unequally distributed (artificial conductivity in the upper mantle will
> be higher than in the lower mantle).
>
> While the above point suggests that the artificial viscosity scheme is
> correctly implemented, there still exist three points for possible
> improvements:
>
>  - As Wolfgang mentioned the parameters that were originally chosen for
> the artificial viscosity scheme were different from what they are today,
> because we noticed that the original smoothing was too weak to stabilize
> oscillations in compositional fields. It is completely possible that the
> original values were sufficient for the temperature (which already has
> natural diffusion), and in hindsight I should have thought of just using
> different parameters for composition and temperature (though that was 6
> years ago, and I was just a 2nd year PhD student at the time I worked on
> modifying the parameters for stabilizing the composition equation). I can
> easily make a change that allows for different parameters for temperature
> and composition, which would allow everyone to test their favorite values,
> without risking oscillations in compositional fields.
>
> -  Even if we can reduce the parameters it is of course possible that as
> Max pointed out the anisotropic nature of the diffusion in SUPG is more
> appropriate / "less wrong" than the isotropic entropy viscosity for our
> problem. As Juliane and Timo pointed out we could reimplement the content
> of https://github.com/geodynamics/aspect/pull/412 and see if SUPG
> performs better for low resolutions. Does anyone know if the "artificial
> viscosity" of SUPG also scales with h^2? Because if it is linear, we might
> implement something that is better for low resolutions, but worse for high
> resolutions.
>
> - As Max mentioned: We should not need a stabilization for pure conduction
> problems (where velocity is 0), and should modify the algorithm accordingly.
>
> So the next steps could be:
>
> 1. Allow for different stabilization parameters for temperature and
> composition, and check which values are still stable.
> 2. Do not stabilize advection/diffusion solutions where the velocity is
> zero (because it is only a diffusion equation).
> 3. Reimplement the SUPG based on
> https://github.com/geodynamics/aspect/pull/412 and see how it performs
> (at low and high resolutions).
>
> Does that summarize our discussion appropriately? I can easily make the
> code adjustments 1 and 2 (they are easy and useful in any case), and could
> also look into creating an initial version of 3 (although it would take a
> bit of time), but I currently do not have the time for much testing of the
> methods, so I would be greatful if someone else could do the testing and
> benchmarking of the methods.
>
> Best,
> Rene
>
>
> On 08/29/2018 09:19 AM, Max Rudolph wrote:
>
> On Tue, Aug 28, 2018 at 8:01 PM Wolfgang Bangerth <bangerth at colostate.edu>
> wrote:
>
>> On 08/28/2018 05:33 PM, Max Rudolph wrote:
>> >  From this, it is very obvious why the solution to the convection
>> problem at
>> > low resolution is very diffusive and also why the interior temperature
>> is much
>> > closer to the surface temperature than to the CMB temperature because
>> the
>> > artificial viscosity is on the order of 20 times larger than the
>> thermal
>> > conductivity near the surface.
>>
>> Would it be easy to verify whether the artificial viscosity ("artificial
>> conductivity") decreases at the expected rate with mesh refinement?
>>
>
> What is the most helpful way for me to show this? Visualization of a
> couple of slices from the 3D conduction model? I tried to get the depth
> average of artificial viscosity but the postprocessor is not implemented.
>
>
>> > For the conduction problem, the default values of the artificial
>> viscosity are
>> > also much larger than the thermal conductivity.
>>
>> I think that's the point worth investigating. Since in this case the
>> velocity
>> is zero, one would expect the artificial viscosity to also be at least
>> quite
>> small. Why is it not?
>>
>
> *Maybe the spherical and 2D annulus geometry models are returning an
> unhelpful length scale, like planetary radius instead of layer depth?*
>
> aspect/source/simulator/entropy_viscosity.cc (starting line 191):
> // If the velocity is 0 we have to assume a sensible velocity to calculate
> // an artificial diffusion. We choose similar to nondimensional
> // formulations: v ~ thermal_diffusivity / length_scale, which cancels
>     // the density and specific heat from the entropy formulation. It seems
>     // surprising at first that only the conductivity remains, but remember
>     // that this actually *is* an additional artificial diffusion.
>     if (std::abs(global_u_infty) < 1e-50)
>       return parameters.stabilization_beta *
>              max_conductivity / geometry_model->length_scale() *
>              cell_diameter;
>
>
>
>>
>> Best
>>   W.
>>
>> --
>> ------------------------------------------------------------------------
>> Wolfgang Bangerth          email:                 bangerth at colostate.edu
>>                             www: http://www.math.colostate.edu/~bangerth/
>>
>>
>
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>
>
> --
> Rene Gassmoellerhttps://gassmoeller.github.io/
>
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